OFFSET
1,2
COMMENTS
The equality sigma_1(n) = Sum{d|n} d defines one partition of sigma_1(n) into distinct divisors of n. This sequence gives the number of partitions of sigma_1(n) into not necessarily distinct divisors of n.
For prime number p, sigma_1(p) = p+1 and there are only two partitions: p and 1+1+1+...+1 (p summands).
EXAMPLE
For n = 4, sigma_1(4) = 7, divisors(4) = {1,2,4} and 7 = 4+2+1 = 4+1+1+1 = 2+2+2+1 = 2+2+1+1+1 = 2+1+1+1+1+1 = 1+1+1+1+1+1+1.
For n = 9, sigma_1(9) = 13, divisors(9) = {1,3,9} and 13 = 9+3+1 = 9+1+1+1+1 = 3+3+3+3+1 = 3+3+3+1+1+1+1 = 3+3+1+1+1+1+1+1+1 = 3+1+1+1+1+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1+1+1.
PROG
(Magma) v:=[1..47];
for u in v do
u, #RestrictedPartitions(SumOfDivisors(u), {d:d in Divisors(u)});
end for;
(Magma)
a:= func< n | #RestrictedPartitions(SumOfDivisors(n), {d:d in Divisors(n)}) >; [ a(n) : n in [1..47] ];
(PARI) numbpartUsing(n, v, mx=#v)=if(n<1, return(n==0)); sum(i=1, mx, numbpartUsing(n-v[i], v, i)) \\ inefficient;
a(n) = numbpartUsing(sigma(n), divisors(n)); \\ after A018818; Michel Marcus, Feb 27 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Marius A. Burtea, Feb 26 2019
STATUS
approved